1. Field of the Invention
The present invention relates to a piezoelectric resonator, and more particularly to a piezoelectric resonator capable of suppressing occurrence of spurious components, a filter using the same piezoelectric resonator, and a duplexer using the same piezoelectric resonator.
2. Description of the Background Art
Elements included in an electronic apparatus such as a portable apparatus are required to be reduced in size and weight. For example, a filter for use in a portable apparatus is required to be precisely adjustable for a frequency response as well as to be reduced in size. As an exemplary filter which satisfies the above requirements, a filter using a piezoelectric resonator is known (for example, see Japanese Laid-Open Patent Publication No. 60-68711).
Hereinafter, referring to FIGS. 10A-10D, a conventional piezoelectric resonator will be described. FIG. 10A is a cross-sectional view showing a basic structure of a conventional piezoelectric resonator 500. The piezoelectric resonator 500 is structured by sandwiching a piezoelectric body 501 between upper and lower electrodes 502 and 503. The piezoelectric resonator 500 is mounted on a substrate 505 having a cavity 504 formed therein. The cavity 504 can be formed by using a micromachining method to partially etch the substrate 505 from its back side. The piezoelectric resonator 500 is caused to vibrate in a thickness direction when the upper and lower electrodes 502 and 503 apply electric fields in the thickness direction. Next, an operation of the piezoelectric resonator 500 is described in conjunction with longitudinal vibration in the thickness direction of an infinite plate.
FIG. 10B is a schematic perspective view used for explaining the operation of the conventional piezoelectric resonator 500. In the piezoelectric resonator 500, if electric fields are applied between the upper and lower electrodes 502 and 503, electrical energy is converted into mechanical energy in the piezoelectric body 501. Mechanical vibration is induced in the thickness direction, and the induced vibration expands and contracts in the same direction as that of the electric fields. The piezoelectric resonator 500 generally utilizes resonant vibration in the thickness direction of the piezoelectric body 501, and resonates at a frequency whose ½ wavelength is equal to the thickness of the piezoelectric resonator 500. The cavity 504 shown in FIG. 10A is provided to ensure that the longitudinal vibration occurs in the thickness direction of the piezoelectric body 501.
As shown in FIG. 10D, an equivalent circuit of the piezoelectric resonator 500 has both a series resonance portion and a parallel resonance portion. In the equivalent circuit, the series resonance portion consists of a capacitor C1, an inductor L1, and a resistor R1, and a capacitor C0 is connected in parallel to the series resonance portion. In this circuit configuration, as shown in FIG. 10C, an admittance frequency response of the equivalent circuit is such that the admittance is maximized at a resonance frequency fr, and minimized at an antiresonance frequency fa. Here, the resonance frequency fr and the antiresonance frequency fa are, in the following relationship.fr=1/{2π√(L1×C1)}fa=fr√(1+C1/C0) 
It is known that in the case of applying the piezoelectric resonator 500 as described above to a filter, it is necessary to increase the size of an electrode as much as possible from the viewpoint of impedance match (for example, see Japanese Laid-Open Patent Publication No. 60-142607).
However, if the electrode size is increased, a contact area between the electrode and the substrate is inevitably increased in order to ensure strength, so that spurious components are readily excited. In actuality, the vibration portion of the piezoelectric resonator is partially fixed on the substrate, and therefore does not entirely produce free longitudinal vibration in the thickness direction.
As shown in FIG. 11, vibrating portions are classified into a portion A vibrating with one end fixed and a portion B with two ends that freely vibrate. In the portion A, vibration occurs at a resonance frequency f2, while in the portion B, vibration occurs at a resonance frequency f1 (FIG. 11 shows ideal vibration displacement distribution under the above boundary conditions of the portions A and B). Accordingly, if the electrode size is increased, the piezoelectric resonator is susceptible to, for example, vibration in the portion A, as well as to a fundamental mode (a ½ wavelength mode, the frequency f1) of desired vibration in the thickness direction, and therefore unwanted vibration readily occurs in the vicinity of a main resonance frequency (f1). This means that energy essentially used for excitation of vibration in the piezoelectric body is partially lost due to vibration leakage.
Such unwanted vibration occurs because there is an extremely small difference in resonance frequency between the portion A (the resonance frequency f2) and the portion B (the resonance frequency f1), and vibration leakage in the portion A of the substrate causes excitation of spurious vibration. For example, if resonant vibration occurs in the portion B, the fixed end of the portion A (a contact point 501a) restricts vibration of the portion B, and vibration of the portion A caused by the vibration of the portion B causes spurious components to occur in the vicinity of a resonance frequency of the are B. If the caused unwanted vibration, i.e., a spurious frequency, is present between the resonance frequency fr and the antiresonance frequency fa, a spurious component 130 appears as shown in FIG. 12A.
If a filter is formed by connecting piezoelectric resonators, which produce the spurious component 130, in parallel as shown in FIG. 12B, undesirable pass characteristics appear in a portion 140 of a passband as shown in FIG. 12C. Such pass characteristics lead to degradation of communication quality.